The Compound Interest on a sum of Rs 7290 becomes 1283.75 in 9 months. If the rate of interest is compounded quarterly, then what is the rate of interest?a)12 (1/2)%b)10 (1/3)%c)22 (2/9)%d)20%Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given:
Principal amount (P) = Rs 7290
Compound interest (CI) = Rs 1283.75
Time period (t) = 9 months

To find:
Rate of interest (r)

Solution:
We know that the formula to calculate compound interest is:
CI = P(1 + r/n)^(nt) - P
Where CI is the compound interest, P is the principal amount, r is the rate of interest, n is the number of times interest is compounded in a year, and t is the time period.

In this case, the interest is compounded quarterly, so n = 4.

Substituting the given values into the formula, we have:
1283.75 = 7290(1 + r/4)^(4*9) - 7290

Simplifying the equation, we get:
1283.75 = 7290(1 + r/4)^36 - 7290
Dividing both sides of the equation by 7290, we get:
1283.75/7290 = (1 + r/4)^36 - 1

Now, let's solve for (1 + r/4)^36:
(1 + r/4)^36 = 1283.75/7290 + 1
(1 + r/4)^36 = 0.176 + 1
(1 + r/4)^36 = 1.176

Taking the 36th root on both sides of the equation, we have:
1 + r/4 = (1.176)^(1/36)

Subtracting 1 from both sides of the equation, we get:
r/4 = (1.176)^(1/36) - 1

Multiplying both sides of the equation by 4, we get:
r = 4((1.176)^(1/36) - 1)

Calculating the value of (1.176)^(1/36), we find:
(1.176)^(1/36) ≈ 1.002021

Substituting this value into the equation, we have:
r = 4(1.002021 - 1)
r = 4(0.002021)
r ≈ 0.008084

Converting the decimal to a percentage, we find:
r ≈ 0.8084%

Therefore, the rate of interest is approximately 0.8084%, which is closest to option C) 22 (2/9)%.

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